How can my shrinkwrapped clothes stay on top of my character's skin when I animate?
I think you have problems with Weights. 
You can try transfer weights from character body to clothes: https://www.youtube.com/watch?v=stem2OPRAyA
About beard:
In object mode select beard, then armature. Press Ctrl + P and choose "Armature deform - With empty groups".
Then choose a beard, open edit mode. Select the beard and open data tab (on the left side of screen. It has the same label at your version of program. Find Vertex Group which is responsible for the head, choose Weight = 1 and click Assign.